A -graph is a connected
graph such that any two vertices have 0 or 2 common neighbors. -graphs are regular, and
the numbers of -graphs
with vertex degree 0, 1, 2, ... are given by 1, 1, 1, 2, 3, 8, 24, 96, 302, ... (OEIS
A202592; Brouwer).

A subset of -graphs
are implemented in the Wolfram Language
as GraphData["ZeroTwoBipartite", d, k]
and GraphData["ZeroTwoNonBipartite", d, k].

Classes of graphs that are -graphs
include the hypercube and folded
cube graphs. Particular named -graphs are summarized in the following table, ordered
by vertex degree, and some of which are illustrated above.

Brouwer considers the unique 20-vertex -graph, denoted -noncayley transitive graph above, which can be constructed
by letting the vertices be the ordered pairs of distinct elements from the 5-set , where is adjacent to whenever , ,
are distinct and is adjacent to when , ,
, are distinct, so that for some , and the permutation that maps to is an even permutation. Equivalently, it can be
constructed by letting the vertices be the 20 vertices of the dodecahedron, choosing
a fixed partition of the dodecahedron into five tetrahedra, and letting two vertices
be adjacent whenever they either lie in a common tetrahedron, or are joined by an
edge of the dodecahedron.